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Here we have another example of the
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Concrete-Pictorial-Abstract approach.
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Again in the max math book but now for
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addition of two and three digit numbers.
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As you can see over here, you can see
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that there is a concrete aspect which are
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these colored pencils. There are 14
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colored pens and probably every
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classroom has a lot of colored pencils
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around so if for example you would take
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25 of them and then say Tya has 14
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colored pencils. How many colored pencils
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do they have altogether? Then I think
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it's quite clear that you've got a
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concrete example where you're actually
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adding two numbers together. We can make
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this example more pictorial so, like here
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you can see that these bars are being
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used to actually represent the
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quantities, hence colored pencils, 25 of
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them. Tya's colored pencils 14 of them, and
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they can be brought together. And this of
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course can be done in several ways but
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in this case, sort of, the tens have been
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brought together as you can see over
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here. Tens have been brought together and
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then the units together making 25 plus
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14 the same as 30 plus 9 and that
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together makes 39. So here you can really
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see how it goes from the concrete to
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pictorial to abstract which is the sum
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that you wanted to finish with. And in
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this section of the book, the tasks
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continue to be exactly like that.
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Find, for example, 25 plus 44 using
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counting blocks and the column method.
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So here again you can see that there are
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examples that are very pictorial and the
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way that the units and the tens. The tens
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and the ones, have been organized, are
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used to actually
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go to a more abstract representation.
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You could even say that this, because of
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all the circles around it that this also
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has some pictorial elements, but
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hopefully it's quite clear that this
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'scaffolding' which also is a term that
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Bruner coined and perhaps you remember
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that the Concrete-Pictorial-Abstract
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approach originates from a lot of
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principles that Jerome Bruner formulated.
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Here you can see it really in action. You
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go from something concrete to pictorial
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and then the abstract. The [PDF] file has some
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more of these examples, so here is the
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third page, where you can actually see
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that you can go to hundreds as well and
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of course at a certain point it will
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become quite difficult to actually
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represent these quantities with cubes or
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anything else, but hopefully it's quite
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clear that then students and pupils can
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actually use the idea behind it and then
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at a certain point, they might not
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need the representation anymore.
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It is a fallback, you could say, a 'plan B'
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that is always there and that can lead
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to further secure understanding and
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skill and an arithmetic skill to make
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these tasks. And then the chapter
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continues, the section continues with
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some more examples and I really advise you
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to try and make these examples yourself.
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The solutions will also be posted in the
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course, as well of course.